The theory of statistical comparison was formulated (chiefly by David Blackwell in the 1950s) in order to extend the theory of majorization to objects beyond probability distributions, like multivariate statistical models and stochastic transitions, and has played an important role in mathematical statistics ever since. The central concept in statistical comparison is the so-called "information ordering," according to which information need not always be a totally ordered quantity, but often takes on a multi-faceted form whose content may vary depending on its use.

Group representations are ubiquitous in quantum information theory. Many important states or channels are invariant under particular symmetries: for example depolarizing channels, Werner states, isotropic states, GHZ states. Accordingly, computations involving those objects can be simplified by invoking the symmetries of the problem. For that purpose, we need to know which irreducible representations appear in the problem, and how.

Recently, a new formulation of quantum mechanics was suggested which is based on the evolution of classical particles, provided with a sign, rather than standard wave functions. This allows several advantages over other approaches: from a theoretical perspective, it offers a more intuitive framework while, from a numerical point of view, it allows the simulation of complex systems with relatively small computational resources. In this talk, I will first go through the tenets of this new approach.

In this talk I will set out two new contributions to the study of operational tasks in a relativistic quantum setting. First, I will present a generalisation of the task known as ‘summoning,’ in which an unknown quantum state is supplied to an agent and must be returned at a specified point when a corresponding call is made. I will show that when this task is generalised to allow for more than one call to be made, an apparent paradox arises: the extra freedom makes it strictly harder to complete the task.

Recent advances in scaling photonics for universal quantum computation spotlight the need for a thorough understanding of practicalities such as distinguishability in multimode quantum interference. Rather than the usual second quantized approach, we can bring quantum information concepts to bear in first quantization. Distinguishability can then be modelled as entanglement between photonic degrees of freedom, where loss of interference is caused by decoherence due to correlations with an environment carried by the particles themselves. This shows that multiparticle, multimode Fock state

What does it mean for quantum state to be genuinely fully multipartite? Some would say, whenever the state cannot be decomposed as a mixture of states each of which has no entanglement across some partition. I'll argue that this partition-centric thinking is ill-suited for the task of assessing the connectivity of the network required to realize the state.

Violations of Bell inequalities have traditionally been used to refute a local-realistic description of the world. Not surprisingly, under the assumption that the world is quantum, they can be used to certify quantum devices. What is surprising is that in some cases this characterisation turns out to be (almost) complete, i.e.~we can determine (almost) everything about the devices and this phenomenon is known as self-testing of quantum systems.

Studying the usefulness of resources can be formalized via the framework of a resource theory. However, the complete answer to the question whether a certain resource is more useful than another one is often hard to find in many of the numerous applications of the framework. Approximate answers can be found by identifying so-called monotones—measures of "resourcefulness". I will present several generic constructions of monotones, of which many monotones known in the literature are concrete examples of.